342 * On Musical Scales. [July, 
Re and La (737 —170) = 567, differing from a Vth by a comma, 
while the former, which makes this interval a good Vth, gives with 
Sol a IVth (585—172=433) equally erroneous. Again, 170 
assumed for Re gives a good Minor Third (1,170 — 907 =263) with 
Si below it, but a defective one (415 —170=245=263 —18) with 
the Fa above it, while if 152 be used the result is simply reversed. 
Thus it appears that for Re two distinct values, 170 and 152, are 
equally eligible, thus originating two distinct diatonic scales. The 
former is that commonly received: the latter, by way of distinc- 
tion, we shall speak of as the Co-diatonie Scale, as if complementary 
or correlative to the other. 
If music were always played in one key, there would be no 
occasion for intermediate notes; but the change of key necessitates 
their introduction, and moreover introduces the further question of 
temperament, arising from the fact that no mere repetition, ascend- 
ing or descending, of perfect Fifths or Thirds will lead to an exact 
Octave or Octaves of the note we set out from. The nearest. 
approach in the case of Fifths is that of twelve Fifths (12 x 585= 
7,020=7,000+-20), which exceed seven Octaves by 20, or rather 
more than a Comma.* The Thirds, whether major or minor, are 
still more rebellious, as appears from the equations 3 x 322 =966 
=1000-—34, and 4 x 263 =1052=1000 +52. By combining 
Fifths and Thirds, however, a much nearer approach may be made. 
Thus 2x 585-+15 x 322 (2V+15III)=6,000, which shows that 
tuning upwards from Re (in the ordinary diatonic scale, where Re 
=170), fifteen perfect Thirds will lead up precisely to the sixth 
Octave above Do. Again, we have 322+8.x585 (III+8V)= 
5,002 = 5,000 + 2, which shows that tuning upwards from Mi eight 
perfect Fifths brmgs us within the ninth part of a Comma to an 
Octave of Do—a difference perfectly inappretiable. The former of 
these coincidences is of no value in musical theory; the latter, how- 
ever, as will appear hereafter, affords the basis of a chromatic scale 
(¢.e. a scale in which the Octave is divided into twelve intervals, 
designated, whether equal or unequal, as “ semitones”) so very 
slightly tempered that, practically speaking, it may be regarded as 
perfect in every key but one. 
Dismissing, however, the subject of temperament, we shall now 
proceed to inquire (for the first time, I believe, on any distinct 
general principle) how the sharps and flats of the scale can be 
inserted on an instfument like the pianoforte admitting only of 
twelve keys on its finger-board within the compass of an Octave, so 
as to allow of transposition into all keys, both natural and sharp or 
flat, with the least possible deviation from a perfect scale in any one 
key, and with the greatest number of attainable perfect harmonics 
* This interval is sometimes called a Pythagorean comma. 
